Integrand size = 28, antiderivative size = 229 \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {11 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {2} a^{5/2} f}+\frac {7 c^4 \tan (e+f x)}{2 a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac {c^4 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}} \]
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Time = 0.36 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3989, 3972, 481, 592, 596, 536, 209} \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{5/2} f}-\frac {11 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {2} a^{5/2} f}+\frac {7 c^4 \tan (e+f x)}{2 a^2 f \sqrt {a \sec (e+f x)+a}}-\frac {c^4 \sin ^2(e+f x) \tan ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )}{4 f (a \sec (e+f x)+a)^{5/2}}-\frac {c^4 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{4 a f (a \sec (e+f x)+a)^{3/2}} \]
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Rule 209
Rule 481
Rule 536
Rule 592
Rule 596
Rule 3972
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \left (a^4 c^4\right ) \int \frac {\tan ^8(e+f x)}{(a+a \sec (e+f x))^{13/2}} \, dx \\ & = -\frac {\left (2 a^2 c^4\right ) \text {Subst}\left (\int \frac {x^8}{\left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f} \\ & = -\frac {c^4 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {c^4 \text {Subst}\left (\int \frac {x^4 \left (10+6 a x^2\right )}{\left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{2 f} \\ & = -\frac {c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac {c^4 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}+\frac {c^4 \text {Subst}\left (\int \frac {x^2 \left (-6 a-14 a^2 x^2\right )}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{4 a^2 f} \\ & = \frac {7 c^4 \tan (e+f x)}{2 a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac {c^4 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {c^4 \text {Subst}\left (\int \frac {-28 a^2-36 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{4 a^4 f} \\ & = \frac {7 c^4 \tan (e+f x)}{2 a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac {c^4 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {\left (2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 f}+\frac {\left (11 c^4\right ) \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 f} \\ & = \frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {11 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {2} a^{5/2} f}+\frac {7 c^4 \tan (e+f x)}{2 a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac {c^4 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}} \\ \end{align*}
Time = 5.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.72 \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {c^4 \cot \left (\frac {1}{2} (e+f x)\right ) \left ((-4+19 \cos (e+f x)-12 \cos (2 (e+f x))-3 \cos (3 (e+f x))) \sec ^4\left (\frac {1}{2} (e+f x)\right )+32 \arctan \left (\sqrt {-1+\sec (e+f x)}\right ) \cos (e+f x) \sqrt {-1+\sec (e+f x)}-88 \sqrt {2} \arctan \left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cos (e+f x) \sqrt {-1+\sec (e+f x)}\right ) \sec (e+f x)}{16 a^2 f \sqrt {a (1+\sec (e+f x))}} \]
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Time = 6.13 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {c^{4} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (-2 \left (1-\cos \left (f x +e \right )\right )^{5} \csc \left (f x +e \right )^{5}+2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}-\left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-11 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}+7 \csc \left (f x +e \right )-7 \cot \left (f x +e \right )\right )}{2 a^{3} f}\) | \(246\) |
parts | \(\text {Expression too large to display}\) | \(1008\) |
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Time = 1.39 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.86 \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [-\frac {11 \, \sqrt {2} {\left (c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + 3 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 4 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + 3 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) - 4 \, {\left (3 \, c^{4} \cos \left (f x + e\right )^{2} + 9 \, c^{4} \cos \left (f x + e\right ) + 2 \, c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{4 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}, \frac {11 \, \sqrt {2} {\left (c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + 3 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 4 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + 3 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) + 2 \, {\left (3 \, c^{4} \cos \left (f x + e\right )^{2} + 9 \, c^{4} \cos \left (f x + e\right ) + 2 \, c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{2 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}\right ] \]
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\[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=c^{4} \left (\int \left (- \frac {4 \sec {\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {6 \sec ^{2}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {4 \sec ^{3}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \frac {1}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx\right ) \]
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Timed out. \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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